4.3.2 Hence deduce the minimum value of \( \sin x+\cos x \) and give the corresponding values of \( x \), for \( x \in\left[-360^{\circ} ; 360^{\circ}\right] \). TOTAL:

To find the minimum value of \( \sin x + \cos x \), we can start by rewriting the expression in a more manageable form. 1. **Rewrite the expression**: We can express \( \sin x + \cos x \) in terms of a single sine function. We know that: \[ \sin x + \cos x = \sqrt{2} \left( \sin x \cdot \frac{1}{\sqrt{2}} + \cos x \cdot \frac{1}{\sqrt{2}} \right) \] This can be rewritten using the sine addition formula: \[ \sin x + \cos x = \sqrt{2} \sin\left(x + 45^\circ\right) \] 2. **Determine the range**: The sine function, \( \sin\left(x + 45^\circ\right) \), has a range of \([-1, 1]\). Therefore, the expression \( \sqrt{2} \sin\left(x + 45^\circ\right) \) will have a range of: \[ [-\sqrt{2}, \sqrt{2}] \] 3. **Find the minimum value**: The minimum value of \( \sin x + \cos x \) is thus: \[ -\sqrt{2} \] 4. **Find corresponding values of \( x \)**: To find the values of \( x \) for which \( \sin\left(x + 45^\circ\right) = -1 \): \[ x + 45^\circ = 270^\circ + k \cdot 360^\circ \quad (k \in \mathbb{Z}) \] This simplifies to: \[ x = 225^\circ + k \cdot 360^\circ \] Now, we need to find \( x \) within the interval \([-360^\circ, 360^\circ]\): - For \( k = 0 \): \( x = 225^\circ \) - For \( k = -1 \): \( x = 225^\circ - 360^\circ = -135^\circ \) Thus, the corresponding values of \( x \) are \( 225^\circ \) and \( -135^\circ \). ### Summary: - The minimum value of \( \sin x + \cos x \) is \( -\sqrt{2} \). - The corresponding values of \( x \) in the interval \([-360^\circ, 360^\circ]\) are \( 225^\circ \) and \( -135^\circ \).